# Full text of "Treatise On Analysis Vol-Ii"

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```392       XV   NORMED ALGEBRAS AND SPECTRAL THEORY

restriction of T to En admits a totalizing vector an and such that, if jun is the
positive measure on K corresponding to an (15.10.1), then un+1 is a measure with
base /j,n (13.13)/0r each n.

We begin with a decomposition of E as the Hilbert sum of any sequence
(Fn) as described above. Let bn be a totalizing vector for the restriction of Tto
Fn , and let vn be the corresponding positive measure on K. Since the measure
vn is determined only up to equivalence (15.10.7), we may multiply it by a
strictly positive constant so as to ensure that the series of norms ||vj| con-
verges.

By induction on n ^ 2 we define two sequences (v^), (vj[) of positive
measures on K, as follows. The measures v'2 and V2 are those which appear in
the Lebesgue canonical decomposition of v2 relative to vx, V2 being a measure
with base vl9 and v£ disjoint from v1 (13.8.4). For k > 2, the measures v*
and v£ are likewise such that vk = v'k + v'k , where v'k is a measure with base
(vx + v*2 + ---- h v£_ t) and vjj is disjoint from v1 -h v£ + • • • 4- vJL r To each
of these decompositions there corresponds a partition of K consisting of two
universally measurable sets Ek and Bk such that v* is concentrated on Wk and
v'i is concentrated on Ek . Let Fk = Ffk © Fk be the corresponding decomposi-
tion of Fk as a Hilbert sum of mutually orthogonal subspaces. If Fk is identified
with L£(K, vk) (15.10.1) then Fk andF£ admit b'k = (p^ and Z>£ = <pB«fc as totaliz-
ing vectors, respectively (15.10.6). The sequence of measures vx + v£ H- • • • + v£
is increasing, and the norms of these measures are bounded above by

[|v

j|. Hence (13.4.4) this sequence has a least upper bound fa in MR(K),

and fa is also the vague limit of the sequence. The preceding construction
allows us to assume that, if vt is concentrated on El9 then the sets Bl9 E2 , . . . ,
B£, . . . are pairwise disjoint, and vk is identified with cpB»k • ulf If Ej is the
Hilbert sum of Fj and the F£ for k^.2, then E is the Hilbert sum of Ej
and the F£ (k *z 2). The subspace E1 is identified with Lc(K, fa) ; since
H^'H2 = Vjk(B£) ^ llvjtll, the series ^ + b"2 + - - - + b'i + • • - converges in E1?
and its sum al is identified with \$Al, where At is the union of Bx and the
B^ (k ^ 2). Clearly ^ is a totalizing vector for the restriction of T to Ex.

Thus, starting with the given decomposition (Fn) of E, we have constructed
a decomposition of E into Ex and the F£ , where for each k ^ 2 the measure
v'k corresponding to F£ is a measure with base fa. Repeating the construction,
we define for each n a decomposition of E as the Hilbert sum of subspaces
E19 . . . , En , F^i , . . . , Fjjjk , . . . , all stable under T9 such that the restriction of
T to each subspace admits a totalizing vector, and .such that if the corre-
sponding measure is uk for Efc (1 ^ k :g n) and v^k for F^, then p,k+l is a
measure with base uk for 1 <; k <* « — 1, and vj!j!k is a measure with base /*„
for all /: ^ 1. To achieve this we have only to apply the previous reasoning to7); consequently u — a + ib is unitary, and a =
```